98:1581-1590, 2007. First published Jul 5, 2007;  doi:10.1152/jn.00577.2007 J NeurophysiolKevin G. Keenan and Francisco J. Valero-Cuevas

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Experimentally Valid Predictions of Muscle Force and EMG in Models ofMotor-Unit Function Are Most Sensitive to Neural Properties

Kevin G. Keenan1,2 and Francisco J. Valero-Cuevas1,2,3

1Sibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, New York; and 2Department of Biokinesiology &Physical Therapy, and 3Department of Biomedical Engineering, University of Southern California, Los Angeles, California

Submitted 22 May 2007; accepted in final form 4 July 2007

Keenan KG, Valero-Cuevas FJ. Experimentally valid predictions ofmuscle force and EMG in models of motor-unit function are mostsensitive to neural properties. J Neurophysiol 98: 1581–1590, 2007.First published July 5, 2007; doi:10.1152/jn.00577.2007. Computa-tional models of motor-unit populations are the objective implemen-tations of the hypothesized mechanisms by which neural and muscleproperties give rise to electromyograms (EMGs) and force. However,the variability/uncertainty of the parameters used in these models—and how they affect predictions—confounds assessing these hypoth-esized mechanisms. We perform a large-scale computational sensitiv-ity analysis on the state-of-the-art computational model of surfaceEMG, force, and force variability by combining a comprehensivereview of published experimental data with Monte Carlo simulations.To exhaustively explore model performance and robustness, we rannumerous iterative simulations each using a random set of values fornine commonly measured motor neuron and muscle parameters.Parameter values were sampled across their reported experimentalranges. Convergence after 439 simulations found that only 3 simula-tions met our two fitness criteria: approximating the well-establishedexperimental relations for the scaling of EMG amplitude and forcevariability with mean force. An additional 424 simulations preferen-tially sampling the neighborhood of those 3 valid simulations con-verged to reveal 65 additional sets of parameter values for which themodel predictions approximate the experimentally known relations.We find the model is not sensitive to muscle properties but verysensitive to several motor neuron properties—especially peak dis-charge rates and recruitment ranges. Therefore to advance our under-standing of EMG and muscle force, it is critical to evaluate thehypothesized neural mechanisms as implemented in today’s state-of-the-art models of motor unit function. We discuss experimental andanalytical avenues to do so as well as new features that may be addedin future implementations of motor-unit models to improve theirexperimental validity.

I N T R O D U C T I O N

Computational models of motor-unit populations are, bydefinition, the objective implementations of the mechanisms bywhich neural and muscle properties are hypothesized to giverise to electromyograms (EMGs) and force. Unlike regression-based inferences, these so-called forward or predictive models(Baker and Lemon 1998; Fuglevand et al. 1993) are causal inthat they use mathematical equations that simulate the flow ofevents from neural command to electromagnetic signals re-corded at the electrodes or forces produced at the tendons. Thedegree to which a model is able to replicate well-establishedexperimental relations is a test of how well those equationsreflect the underlying mechanisms and a test of our hypotheses

of neuromuscular function. Thus the rigorous study of how tomake these models as experimentally valid as possible iscritical to improve both our understanding of neuromuscularfunction, and our ability to use EMGs to confidently infercommand signals and forces in the neuromuscular system.

Briefly, the EMG signal is the sum of motor-unit potentialsdetected by electrodes and is the most common experimentalmeans to estimate the neural drive to a muscle (Lacquaniti andSoechting 1982; Schieber 1995; Thoroughman and Shadmehr1999; Valero-Cuevas et al. 1998) and muscle force (Inman etal. 1952; Milner-Brown and Stein 1975). Because many factorsinfluence the EMG signal (Basmajian and De Luca 1985), mostresearchers agree that it has important limitations (Day andHulliger 2001; Farina et al. 2004a). Due to these limitationsand the current absence of other approaches to examine inter-actions of entire motor-unit populations, computational modelsare fast becoming the dominant means to establish the mech-anisms relating EMG to muscle force and neural drive (Bakerand Lemon 1998; Fuglevand et al. 1993; Stegeman et al. 2000).Structure-based EMG models (reviewed in McGill 2004;Stegeman et al. 2000) use motor units as their fundamentalbuilding blocks and promise to indicate how to best use EMGto estimate muscle force and the neural drive to muscle in abehaving animal.

Critically assessing these computational models of motorunits is relevant and urgent because in the absence of othercomputational means, these models are routinely used to studyand debate the effects of motor neuron and muscle parametervalues on muscle function. Some current debates center onpeak discharge rates of motor units (Connelly et al. 1999;Enoka and Fuglevand 2001; Johns and Fuglevand 2004),recruitment ranges (De Luca et al. 1982; Moritz et al. 2005),number of motor units (Hamilton et al. 2004; Heckman andEnoka 2004), and range in innervation numbers (Enoka andFuglevand 2001). However, the variability and uncertainty ofthe parameters used in these models confounds assessing howrealistic they are. Given the excessive computational costneeded to model the behavior of an entire motor-unit pool,typically requiring millions of calculations, simulations havepreviously been restricted to examining the influence of inde-pendent changes in single parameters on single outputs (Joneset al. 2002; Keenan et al. 2005; Zhou and Rymer 2004).Validation efforts to date have not addressed multivariateinteractions, the potential confound of an inappropriate modelstructure (Valero-Cuevas 2005), intra- and inter-subject vari-

Address for reprint requests and other correspondence: K. G. Keenan, SibleySchool of Mechanical and Aerospace Engineering, Cornell University, Ithaca,NY 14853 (E-mail: kk375@cornell.edu).

The costs of publication of this article were defrayed in part by the paymentof page charges. The article must therefore be hereby marked “advertisement”in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.

J Neurophysiol 98: 1581–1590, 2007.First published July 5, 2007; doi:10.1152/jn.00577.2007.

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ability in motor-unit properties (Feinstein et al. 1955), nor thefact that many properties across an entire population of motorunits are largely unknown (Enoka and Fuglevand 2001).

We use multivariate Monte Carlo simulations to exhaus-tively and simultaneously explore the sensitivity of modelpredictions to uncertainty and variability in nine of the mostcommonly investigated motor-unit parameters. Motor-unitmodels should, at the very least replicate known relationshipsamong force, force variability, and EMG amplitude and berobust to parameter variability across experimentally identifiedranges for motor neuron and muscle properties. It is welldocumented that both EMG amplitude and force variabilityscale with mean force level (Bigland-Ritchie 1981; Enoka et al.2003; Jones et al. 2002; Milner-Brown and Stein 1975). Thisstudy examined the ability of the most popular motor-unitmodel to replicate experimental EMG/force and force/force-variability relations in the presence of variability and uncer-tainty in parameter values. A preliminary account of part ofthese findings has been published in abstract form (Keenan andValero-Cuevas 2006).

M E T H O D S

We used Monte Carlo simulations to test the ability of the mostpopular motor-unit model available (Fuglevand et al. 1993) to ro-bustly predict well-established relations among EMG, force, and forcevariability when changing the values for motor neuron and muscleparameter values across experimentally identified ranges. Our com-puter simulations are based on the forward model of recruitment of apopulation of motor units and production of muscle force (Fuglevandet al. 1993) with the addition of an EMG model (Farina and Merletti2001) that simulated the surface EMG with a planar volume conductorwith muscle, fat, and skin tissues. Motor-unit potentials and the EMGsignal were computed at 4,096 sample/s and muscle force at 500sample/s. Monte Carlo methods, as their name implies, use exhaustiverepeated simulations where the specific value of each free modelparameter is assigned by random chance from a preassigned range,and the output is computed (Santos and Valero-Cuevas 2006; Valero-Cuevas et al. 2003). Each iteration consists of a simulation that usesone given set of values for the free parameters (see details in thefollowing text). On convergence the results are in the form of therange and variability in the output in response to changes in theparameter values—which is a rigorous evaluation of the robustnessand realism of the structure of the model in relation to the variability/uncertainty in parameter values. “Convergence” in Monte Carlosimulations has the specific meaning that the number of simulationsperformed has sufficed to produce stable estimates of the statisticalparameters of the output distributions. Importantly, convergence cer-tifies that further iterations are no longer useful because they wouldnot alter the results. Thus converged Monte Carlo simulations are acomputationally rigorous means to thoroughly explore large parame-ter spaces (Santos and Valero-Cuevas 2006; Valero-Cuevas et al.2003). The model and Monte Carlo iterations are implemented inMatlab version 7.0 (The Mathworks, Natick, MA).

As described in the extensive literature using this model (Jones etal. 2002; Keenan et al. 2006a; Yao et al. 2000) each iteration of ourMonte Carlo simulations comprised six main sequential steps: 1)determining the recruitment and discharge times of a population ofmotor neurons in response to the level of excitatory drive (we used 6levels for each Monte Carlo simulation); 2) generating motor-unitpotentials from a given number, location, and conduction velocities ofthe muscle fibers for each motor unit; 3) simulating the surface EMGby summing the trains of motor-unit potentials; 4) generating twitchforces from the given number of muscle fibers for each motor unit; 5)implementing the sigmoidal relation between motor-unit force and

discharge rate frequency as a function of the instantaneous dischargerate and contraction time of the unit; and 6) simulating muscle forceby summing motor-unit forces. On convergence, the cumulativeresults from the Monte Carlo simulations provide statistical distribu-tions of average full-wave rectified EMG amplitudes, and the meanand SD of force that result from the reported variability and measure-ment uncertainty in motor-unit parameters (Table 1).

Because the original equations (Farina et al. 2004b; Fuglevand et al.1993) and our implementation (Keenan et al. 2005, 2006a,b) of themodel have previously been published, the following section will onlyreview key components of the model and focus on our novel MonteCarlo implementation.

Motor neuron pool model

The distributions of properties across the motor-unit pool werebased on the size principle (Henneman 1957), and these includedinnervation number, twitch force, recruitment threshold, motor-unitterritory, and conduction velocity of motor-unit potentials. The totalnumber of motor neurons in the pool varied from 150–500 and thenumber of muscle fibers from 50,000–250,000 (Table 1).

Activation of the motor-unit pool is modeled as a ramp-and-holdfunction, with a 1-s ramp increase in excitation to a constant level for10 s. Excitation ranges from 0 to 1, with maximal excitation definedas the level of activation necessary to bring the last recruited motorneuron to its assigned peak discharge rate. The distribution of recruit-ment thresholds for the motor neurons is represented as an exponentialfunction with many low-threshold neurons and progressively fewerhigh-threshold neurons. The range over which recruitment occursvaries from �30 to 80% of maximal excitation (Table 1), consistentwith the wide range found experimentally (De Luca et al. 1982;Moritz et al. 2005). Each motor unit began discharging at 8 pulse/s(pps) once excitation exceeded the assigned recruitment threshold forthat unit, and discharge rates increase linearly with increases inexcitation. The interspike interval for motor-unit discharge is modeledas a random process with a Gaussian probability distribution function,with the coefficient of variation [CV � 100 � (SD/mean) %] of theinterspike intervals fixed for all motor units at 20%. The influence ofthe correlated discharge among motor units (synchronization) was notinvestigated as it has minimal influence on normalized EMG ampli-tudes (Keenan et al. 2005; Zhou and Rymer 2004).

High-threshold motor neurons have been reported to obtain eitherhigher (Gydikov and Kosarov 1974; Moritz et al. 2005), lower (DeLuca et al. 1982), or similar (Erim et al. 1996) peak discharge rateswith respect to low-threshold motor neurons. Given that evidence foronly one discharge rate strategy has not been clearly established invivo, peak discharge rates are assigned for the first- and last-recruitedmotor neuron independently and are drawn from a uniform distribu-tion that varied from 25 to 50 pps (Table 1).

TABLE 1. Model parameters and the ranges over whichthey varied

Properties Parameters Ranges

Motor neuron 1) Number of motor neurons 150–5002) Range in innervation numbers 1–1003) Recruitment range Up to 30–80% maximal

excitationPeak discharge rate:

4) First recruited neuron 25–50 pulses per second5) Last recruited neuron 25–50 pulses per second

Muscle 6) Number of fibers 50,000–250,0007) Fiber length 4–16 cm8) Mean conduction velocity 3–4 m/s9) Conduction velocity spread 0–0.5 m/s (SD)

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Muscle force generation model

Motor unit twitch force is modeled as the impulse response of acritically damped, second-order system (Milner-Brown et al. 1973a).A large number of units produce small forces, whereas relatively fewunits generated large forces. Twitch force is modeled as directlyproportional to the number of fibers assigned to each motor neuron(innervation number), and the range in innervation numbers, calcu-lated by dividing the innervation number of the last-recruited motorunit by the first-recruited motor unit, varies from 1 to 100� (Table 1).Variation in the range in innervation numbers, and thus twitch forcesin the model, is simulated because experimental reports using glyco-gen depletion techniques find ranges �2�, whereas estimates ofranges in innervation numbers using twitch and tetanic motor-unitforces, which are highly correlated with innervation number (Bodineet al. 1987; Kanda and Hashizume 1992), can extend �100� (re-viewed in Enoka and Fuglevand 2001). Innervation numbers increaseexponentially from the smallest to the largest motor unit. There is aninverse relation between twitch force and twitch contraction time. Thefirst- and last-recruited unit are assigned a twitch contraction time topeak of 90 and 30 ms, respectively. All motor units follow theestablished sigmoidal relation between motor-unit force and motor-unit discharge rate (Bigland and Lippold 1954; Fuglevand et al. 1993;Kernell et al. 1983). Total force of the muscle is calculated as thelinear summation of all the individual motor-unit forces.

Surface EMG simulations

The planar volume conductor consists of an isotropic fat (3-mmthick) and skin layer (1-mm thick), and an anisotropic muscle layer.Each tissue layer is homogeneous and the conductivity properties ofeach layer are similar to those reported by Farina et al. (2004b). Thecross-section of the muscle is simulated as circular with a radiusproportional to the assigned number of muscle fibers (Table 1).Average fiber length varies from 4 to 16 cm (Table 1), and the centerof the innervation zones is located at 50% of the length of the fibersand in the same transverse location within the volume conductor. Theend-plate and insertion of each fiber into the tendons varies randomly(uniform distribution) over a range of 5 mm. The electrodes aresimulated as circular (4-mm diam) and arranged in bipolar pairs(10-mm interelectrode distance) along the direction of the musclefibers. The electrodes are located halfway between the innervationzone and the distal tendon and are placed directly over the center of

the muscle in the radial direction. The simulation of the intracellularaction potential is based on the analytical description of Rosenfalk(1969). Motor units have mean muscle fiber conduction velocities thatvary from 3 to 4 m/s and have a spread (SD) that varies from 0.0 to0.5 m/s (Farina et al. 2000) (Table 1), with the smallest motor unitsassigned the slowest conduction velocities (Andreassen and Arendt-Nielsen 1987). The motor-unit potential comprises the sum of theaction potentials of the muscle fibers belonging to the motor unit andthe surface EMG is simulated by summing the trains of motor-unitpotentials.

The distribution of the motor units within the muscle (Johnson et al.1973; Milner-Brown and Stein 1975) is determined by randomlyselecting the x-y coordinates corresponding to the center of eachmotor-unit territory within the circular cross-section of the muscle.The fibers of a motor unit are randomly scattered in the motor-unitterritory (Stålberg and Antoni 1980) with a density of 20 fibers/mm2

(Armstrong et al. 1988) and interdigitated with fibers belonging tomany other units. The territories of the largest motor units, thereforeare greater than those of the smallest motor units (Bodine et al. 1988).To restrict the distribution of the fibers of a motor unit (Bodine et al.1988; Stålberg and Antoni 1980), motor-unit territories are modeledas circular (Buchthal et al. 1959). The radius of the motor-unitterritory is based on the density of 20 fibers/mm2 and the assignedinnervation number of the motor unit. However, when a portion of themotor-unit territory is constrained by the muscle boundary, the radiusof the territory of the unit is augmented to accommodate the requirednumber of motor-unit fibers within the new territory while maintain-ing fiber density.

Monte Carlo simulations

We investigated how changes in five motor neuron and four muscleparameter values (Table 1) affect the predicted relations among EMG,force, and force variability. These nine parameters are specificallychosen because they are intensively investigated in animal and humanexperiments and also because the influence of these properties onEMG and force production is the focus of the majority of studies usingmotor-unit models (e.g., Fuglevand et al. 1993; Jones et al. 2002;Keenan et al. 2005). Three muscle properties (mean and variability inmotor-unit conduction velocities, and fiber length) were varied tospecifically test their influence on the surface EMG. The Monte Carloapproach is demonstrated in Fig. 1 for the generation of the surface

FIG. 1. We began a multi-stage validation of a state-of-the-art computational model of normalized electromyographic (EMG) amplitude by performing MonteCarlo simulations to determine the sensitivity of predicted EMG amplitude to 5 motor neuron and 4 muscle properties (Table 1). Iterative simulations sampleddifferent random combinations of property values across the full activation range. For each iteration, an activation pattern, representing the recruitment and ratecoding of a population of motor units (e.g., 150), is computed based on the parameter values chosen from prior distributions. Surface-detected motor-unitpotentials are generated for the given anatomical factors and detection scheme (details in METHODS), and the surface EMG is formed by summing the trains ofmotor-unit potentials at 6 excitation levels. Model outputs are in the form of EMG amplitude distributions. To exhaustively search the parameter space, 1,095iterations were performed to satisfy convergence criteria (�2% change in running mean and coefficient of variation of EMG amplitude for the last 20�% ofthe iterations). The same procedure was used to simulate force and force variability. MU, motor-unit label; pps, pulse/s.

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EMG with a population of 150 motor units. For each Monte Carloiteration, we defined a set of parameter values by sampling at randomfrom uniform distributions for each of the nine parameters bounded bythe experimental range of values commonly reported for each (Table1). For each iteration, an activation pattern (motor-unit recruitmentand discharge rates) was simulated at six levels of excitation (arrowsin Fig. 1) based on the motor neuron property values drawn. For theanatomical factors previously described in METHODS (Fig. 1), and themotor neuron and muscle properties drawn for each iteration, motor-unit potentials are generated for each motor unit, and an EMG signalis simulated at each excitation level by summing all trains of motor-unit potentials. The same Monte Carlo approach is used to simulatemuscle force.

Data analysis

Force and full-wave rectified EMG are averaged over a 10-swindow during constant excitation, and force variability is quantifiedas the SD of the detrended force. Convergence was declared whenthere was �2% change in the running mean and coefficient ofvariation of EMG amplitude and force for the last 20�% of thesimulations (Santos and Valero-Cuevas 2006; Valero-Cuevas et al.2003). Note that each Monte Carlo iteration itself required conver-gence at every one of the six levels of excitation.

Because EMG is influenced by the distribution of motor-unitterritories within the muscle, published simulations generally involvean arbitrary number (e.g., 20) of random motor-unit locations (Farinaet al. 2002; Lowery et al. 2003), while holding all other parametersfixed (e.g., discharge patterns, conduction velocities, and number offibers for each motor unit). To ensure that this variability did not affectour results, EMGs within each Monte Carlo iteration were alsocomputed iteratively with randomly located motor-unit territoriesuntil EMG amplitudes also met the 2% convergence criteria. Onaverage, 107 such iterations sufficed.

Two fitness criteria were chosen to indicate a reasonable matchbetween simulated and experimental EMG/force and force/force-variability relations. These relations were determined by regressionanalysis done in MATLAB. First, force variability is known to scalewith mean force (Enoka et al. 1999; Moritz et al. 2005; Slifkin andNewell 2000), and Jones and Wolpert (2002) reported that the meanslope of the log-log regression line between SD and mean force was1.05. We chose a slope �0.75 and �1.25 to indicate a log-log linear(slope � 1) scaling consistent with signal-dependent noise (Harris andWolpert 1998). And second, EMG amplitude is known to scale withforce (Bigland-Ritchie 1981; Inman et al. 1952; Milner-Brown andStein 1975), with the EMG/force relation described as either linear ornonlinear—with EMG increasing less rapidly than force at moderatecontraction levels. We chose a slope in the regression line betweenEMG and force of �1.05 as a liberal estimate of a match withexperimental observations so as to not over-constrain the modeloutput a priori. To calculate EMG/force relations, EMG and forcewere normalized with respect to their values at maximal excitation foreach Monte Carlo iteration. Convergence was assessed using outputdistributions of EMG amplitude (Fig. 1) and force, instead of usingoutput distributions of the EMG/force and force/force-variabilityslopes. This resulted in a more rigorous sampling of the parameterspace because more than twice as many Monte Carlo iterations werenecessary to satisfy the convergence criteria on the former than on thelater output distributions.

On convergence, we identified the iterations that met the experi-mental fitness criteria and analyzed two aspects of their associated setsof parameter values. First, we established and compared the ranges ofvalues of each model parameter for which the model producesexperimentally valid simulations. This explicitly identifies the param-eters to which the model is most sensitive. Second, we rank orderedthese sets of valid parameter values based on how well they met theexperimental fitness criteria. This provides specific sets of parameter

values for others to run experimentally valid simulations with differ-ent fitness emphases.

R E S U L T S

The initial Monte Carlo simulations converged after 439iterations, but only three simulations approximate the experi-mental relations for force/force-variability and EMG/force.Figure 2 shows the 439 Monte Carlo iterations plotted againstthe required experimental relations in force/force-variabilityand EMG/force. Horizontal and vertical dashed lines indicateboundaries for the fitness criteria representing an approximatematch with experimental data (Fig. 2C), with quadrant IVcontaining those iterations that match both relations. In gen-eral, the model showed a predominant nonlinear scaling offorce variability (Fig. 2A, green) and EMG (Fig. 2B, red) with

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imental EMG/force and force/force-variability relations for 3 of 439 MonteCarlo iterations. The relation between mean force and the SD of force (A) andEMG (B) is shown for 439 Monte Carlo iterations. Two criteria (- - -, C) wereused to compare the model results with experimental findings: a near linearincrease in force variability with mean force (regression line slope �0.75), anda near linear or less than linear EMG/force relation (regression line slope�1.05). Monte Carlo iterations that satisfied the criterion for EMG/forcerelations are in green, force/force-variability relations are in red, both relationsare in black, and neither relation are in gray. Of 439 Monte Carlo iterations,only 3 satisfied both criteria (A–C - quadrant IV, black). The nominal fitnesspoint (1,1), which represents the linear scaling (slope � 1) of force variabilityand EMG amplitude with mean force, is provided for reference.

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mean force across 6 excitation levels, in contradiction to whatis observed experimentally. Only 3 of the 439 Monte Carloiterations resulted in an approximate match with experimentalrelations (Fig. 2, black). Histograms for the nine motor neuronand muscles properties are depicted in Fig. 3 for the parametervalues that lead to an approximate match with experimentalfindings.

We found a tendency toward a tradeoff between fitnesscriteria where satisfying one fitness criterion was negativelyassociated with satisfying the other. Specifically, the slopes ofthe EMG/force relations were analyzed as a function of theslopes of the force/force-variability relations (Fig. 2C) with afirst-order regression model, and the two were slightly posi-tively correlated (r � 0.38, P � 0.05). This suggests that usinga set of parameters that replicate experimental EMG-forcerelations tended not to replicate the force/force-variabilityrelations. In fact, close inspection of the valid parameter setssuggests that satisfying each fitness criterion involved prefer-entially sampling from different regions of the parameterspace. For example, parameter values for peak discharge rateof the last recruited motor neuron that resulted in valid force/force-variability relations were always �35 pps, while fewervalues �35 pps resulted in valid EMG/force relations (Fig. 4).

We performed a second set of Monte Carlo simulations, nowpreferentially sampling from the parameter ranges identified inFig. 3, to better characterize the subset of the parameter spacethat might lead to simulations compatible with experimentaldata. Specifically for this second set of simulations, the prob-ability of selection from the truncated range bounded byobservations in Fig. 3 was 0.8, whereas the probability ofsampling from the remaining portion of the range was 0.2 (Fig.3, gray region). Convergence was met after 424 iterations forthe second set of simulations (Fig. 5), and 65 iterations met ourfitness criteria. Figure 6 depicts force/force-variability (A) andEMG/force relations (B) for those 65 valid iterations.

Examination of the histograms for the parameter sets (Fig. 7)from the second set of simulations (seen in Figs. 5 and 6) showthat the model is not robust to parameter variability and is mostsensitive to motor neuron properties. This lack of robustness

and greater sensitivity is made evident by the fact that three ofthe nine histograms cover �50% of the experimentally-iden-tified ranges, and all three are motor neuron properties (Fig. 7).Specifically, the parameter sets that met with our experimentalfitness criteria required peak discharge rates for the first- andlast-recruited motor neuron to be �33 pulses/s, and recruit-ment ranges to be �60% maximal excitation. In contrast, eachmuscle property could vary across its full range and stillproduce valid EMG/force and force/force-variability relations.In addition, a final Monte Carlo simulation injecting smalluniformly distributed perturbations (�5%) to the best param-eter set (i.e., that most closely approximated the linear scalingof EMG and force variability with force) led to valid solutionsfor 132 iterations, after converging with 232 iterations (only56.9% success rate). That best parameter set (Fig. 5, blueasterisk) was selected from the 863 simulations shown in Figs.2 and 5. The inset in Fig. 5 depicts the fitness for those 232iterations.

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FIG. 3. Histograms for the 5 motor neu-ron and 4 muscle properties show the param-eter values (■ ) that resulted in the 3 MonteCarlo iterations that satisfied both criteria,thus representing a match with experimentaldata. As only 3 iterations satisfied both cri-teria, a 2nd set of Monte Carlo simulationsfurther examined the subset of parametervalues necessary to replicate experimentaldata. Specifically, we preferentially sampledfrom the range in parameter values identifiedfrom this first set of simulations, with theprobability of selecting from the truncatedparameter ranges being 0.8 and the probabil-ity of sampling from the remaining portionwas 0.2 (shown in gray).

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FIG. 4. Histograms for the peak discharge rate of the last recruited neuronshow the parameter values that satisfy the criteria representing a match withexperimental EMG/force (green) or force/force-variability (red) relations.Identifying the intersection for both sets of parameter values, satisfying bothcriteria necessarily involves sampling from a narrow subset of all feasibleparameter values. This is demonstrated in Fig. 3 from the histogram for thepeak discharge rate of the last recruited neuron that satisfies both fitnesscriteria.

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Table 2 presents the 50 parameter sets from all 1,095 (i.e.,439 � 424 � 232) Monte Carlo iterations that best approxi-mate experimental findings, ranked in order of their proximityto the nominal fitness point (1,1), which is the reproduction ofthe commonly reported linear scaling (slope � 1) of forcevariability and EMG amplitude with mean force. As the slopesof simulated EMG/force relations were always �1 while sat-isfying both fitness criteria (Figs. 2C and 5), it was not possibleto assess experimentally observed nonlinear (slope �1) EMG/force relations.

D I S C U S S I O N

Most importantly, our work systematically evaluates themechanisms by which neural and muscle properties are hy-pothesized to give rise to EMG and force. The equations in themodel are, in their essence, the objective implementation ofthese hypotheses. Finding that the model is most sensitive toneural properties indicates that to advance our understanding ofEMG and muscle force, it is critical to evaluate and potentiallyrevise the equations representing the neural mechanisms in thispopular model. As discussed in the following text in detail, itis likely critical to re-evaluate how recruitment ranges and peakdischarge rates affect EMG and muscle force. In addition, wefind that the model, as implemented in today’s literature, tendsto trade-off between the two well-established experimentalrelations we used as validation criteria: the EMG/force andforce/force-variability relations. Nevertheless, we found pa-rameter sets for which the model approximates both theseexperimental relations. We begin by discussing limitations andfinish by suggesting experimental and computational means toimprove our understanding and representation of neural mech-anisms in the model, as well as additional features that may beadded to the model, to make it as experimentally valid aspossible.

Our results should be interpreted with respect to the limita-tions of the modeling approach, which we nevertheless believedo not affect the validity of our conclusions. All Monte Carlosimulations are limited by assuming parameter independence,resulting in parameter sets and output distributions that may bebroader than are likely to exist in reality. However, when thereis a lack of experimental evidence describing whether and howthe free parameters co-vary (Table 1), it is necessary to assumeparameter independence to obtain an unbiased sense of allpossible model outputs. Additionally, this first large-scalecomputational study did not vary all possible parameters in themodel because the number of simulations to convergencegrows exponentially with the number of free parameters.Rather we focused on those parameters that are most com-monly investigated and segregate into intuitively neural versusmuscle properties. The other possible parameters were retainedas fixed to values most commonly used in the literature(Fuglevand et al. 1993; Keenan et al. 2005). Our results nowenable and guide the future exploration of the hypothesesimplemented in the model by the systematic expansion of thenumber of free parameters to determine if a better match withexperimental data are possible. As stated in the preceding text,the choice of our fitness criteria was motivated by a desire toapproximate the well-established experimental relations for thescaling of EMG amplitude and force variability with meanforce. A limitation of our fitness criteria could be that the slopeof the force/force-variability relationship on a log-log scaletends to mitigate the impact of force variability at low activa-tion levels (Laidlaw et al. 2000). Future studies could certainly

103 104 105101

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103

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(au

)

0 20 40 60 80 1000

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(% m

axim

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)

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FIG. 6. The ability of the motor-unit model to match experimental EMG/force and force/force-variability relations is shown. The motor-unit modelreplicated EMG/force and force/force-variability relations for 65 of 424 MonteCarlo iterations. Note the near linear scaling of force variability (top) and EMG(bottom) with mean force, and the inability of the model to replicate the lessthan linear increase in EMG with force commonly reported. Au, arbitrary units.

*

0 0.25 0.5 0.75 1 1.25

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1

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/force (slo

pe)

*+

FIG. 5. By preferentially sampling from the parameter ranges identified inthe 1st set of simulations (Fig. 3), the motor-unit model replicated EMG/forceand force/force-variability relations for 65 of 424 Monte Carlo iterations.Convergence criteria were met after 424 iterations, although only 65 iterations(�15%) satisfied both fitness criteria (quadrant IV, black). After a total of 863simulations, no trials reproduced the nominal fitness point (1,1). The bestparameter set (i.e., that most closely approximated the linear scaling of EMGand force variability with force) is shown with a blue asterisk. Inset: results fora final set of Monte Carlo simulations, which involved small perturbations(�5%) to the values for that best parameter set. After converging with 232iterations, 132 were valid solutions (56.9% success rate).

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refine or change fitness criteria. Last, alternative motor-unitmodel structures have been reported (e.g., simulating the syn-aptic inputs onto the motor neuron pool) (Baker and Lemon1998; Lowery and Erim 2005). However, those models likelyshare similar hypothesized mechanisms and may therefore besusceptible to similar limitations as those presented here.Nonetheless future work can perform similarly thorough ex-ploration of their parameter spaces.

The dominant sensitivity of the model to motor neuronparameter values indicates the equations representing neuralfunction may require refinement. This conclusion is reachedusing two lines of reasoning. First, when the model is mostsensitive to a few parameters (i.e., defined here as spanning anarrow range of values for a group of valid simulations, Fig. 7),it means that the equations using those parameters are mostcritical to producing those valid solutions.1 In this case, we findthat the equations using parameters associated with neuralproperties of the motor-unit populations are most influential tomodel fitness. Second, the model, as implemented in theliterature, needs refinement because it does not produce simu-lations whose fitness is comfortably within the experimentallyestablished ranges for any combination of parameter values[i.e., the distribution of valid simulations (Fig. 5) does notinclude the nominal fitness point (1,1)]. The reason this can beconcluded is that a converged Monte Carlo simulation indi-cates that further sampling of the parameter space will notproduce qualitatively different results (Santos and Valero-Cuevas 2006; Valero-Cuevas 2005); thus the nominal fitnesspoint (1,1) will not be the mode of the distribution of validsimulations even if the model is run for an infinite number ofiterations. Furthermore, while satisfying both fitness criteria,EMG amplitude was never less than force at intermediatelevels of activation (Fig. 6B) as is sometimes observed exper-imentally (Bigland-Ritchie 1981; Lawrence and De Luca1983). Therefore the model warrants improvement, and thisimprovement may be brought about at this point by refining the

equations in the model representing the neural mechanisms ofmotor-unit function.

The narrow ranges of valid motor neuron parameter values(Fig. 7), which resulted in part by the trade-off betweenEMG/force and force/force-variability relations (Fig. 2C), aredifficult to reconcile with current debates in the literature overthose parameters. Specifically, peak discharge rates for the firstand last recruited motor unit had to be �33 pps, and the rangeover which recruitment occurred had to be �65% maximalexcitation to reproduce experimentally valid solutions. In con-trast, experimental studies report peak discharge rates �50 pps(Bellemare et al. 1983; Kamen et al. 1995) and recruitmentcomplete by �50% maximal force (De Luca et al. 1982;Milner-Brown et al. 1973b). Also, the need to preferentiallysample different parameter values (e.g., peak discharge rates—Fig. 4) to independently satisfy each fitness criterion mustnecessarily be an inherent artifact of the model’s structure.This disparity between parameter values necessary for themodel to approximate realistic simulations versus parametervalues known to exist experimentally likely limits the utilitythis model structure to study muscle function when assumingpeak discharge rates �35 pps (Jones et al. 2002; Zhou andRymer 2004) and recruitment ranges �50% maximal excita-tion (Fuglevand et al. 1993; Keenan et al. 2005). However,having identified these narrow parameter ranges is particularlyuseful because they identify which aspects of the model struc-ture (i.e., equations hypothesized to represent neural mecha-nisms of motor-unit function) would benefit most from sys-tematic evaluation. For example, peak discharge rates definemaximum excitation in the model (see METHODS), determinemotor-unit recruitment and discharge rate at each level ofactivation, and establish when discharge rates saturate withincreases in activation of the motor-unit pool.

Importantly, our exhaustive search of the nine-dimensionalparameter space using 1,095 Monte Carlo iterations did iden-tify sets of parameter values that resulted in an acceptablematch with the experimental observations describing EMG/force and force/force-variability relations (Figs. 2C and 5,quadrant IV, Table 2). Although experiments and motor-unit

1 The converse is also true: when a larger range of values is used to producevalid solutions, it means that the equations using those parameters are lesscritical to producing valid solutions.

150 200 250 300 350 400 450 5000

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Recruitment range (% maximal excitation)Co

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Motor Neuron Properties Muscle Properties

1

FIG. 7. Histograms for the 5 motor neu-ron and 4 muscle properties depict the pa-rameter subsets necessary to match experi-mental data. The model was sensitive tomotor neuron properties, with no iterationsthat satisfied both fitness criteria when peakdischarge rates for the first- and last-re-cruited neuron were �33 pps and recruit-ment was complete by 64% maximal excita-tion. Upper and lower bounds for valid pa-rameter ranges are shown (- - -).

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models have consistently characterized relations between bothforce and force variability (Jones et al. 2002; Laidlaw et al.2000; Moritz et al. 2005) and EMG (Bigland-Ritchie 1981;Inman et al. 1952; Milner-Brown and Stein 1975), to ourknowledge no previous study has evaluated motor-unit modelsby comparing their predictions against both of these well-established relations. For example, previous univariate ap-proaches addressed which factors influence force/force-vari-

ability relations, including: discharge rate variability (Moritz etal. 2005), range in twitch amplitudes (Jones et al. 2002), andnumber of motor units (Hamilton et al. 2004). In addition,previous univariate approaches addressed which factors influ-ence EMG/force relations, including motor-unit synchroniza-tion (Yao et al. 2000), electrical and mechanical properties atthe individual motor-unit level (Zhou and Rymer 2004), am-plitude cancellation (Day and Hulliger 2001; Keenan et al.

TABLE 2. Parameter sets for the 50 experimentally valid simulations that best matched EMG/force and force/force-variability relations

TrialSlope Force/

Force-SDSlope

EMG/force

Parameter Values

1 2 3 4 5 6 7 8 9

nMN rIN Rr PDR1 PDRn nMF FL CV CVSD

1 0.768 1.015 383 48.3 79.4 28.5 29.8 70182 5.88 3.28 0.0792 0.847 1.017 398 52.7 77.5 26.6 28.6 69983 5.68 3.35 0.0793 0.828 1.017 384 51.4 77.3 26.7 29.1 71609 5.57 3.31 0.0784 0.833 1.018 383 50.4 79.7 26.6 30.3 71285 5.90 3.31 0.0825 0.761 1.021 382 50.2 78.8 28.7 29.8 72221 5.79 3.37 0.0806 0.760 1.021 383 49.6 79.9 29.1 30.0 69322 6.07 3.31 0.0817 0.773 1.021 372 52.6 79.2 28.0 30.0 74469 5.67 3.43 0.0808 0.610 1.022 364 46.3 79.6 27.5 30.0 69068 5.99 3.44 0.0849 0.789 1.022 379 50.2 77.8 27.1 29.0 69510 5.62 3.42 0.085

10 0.773 1.022 399 52.4 77.7 28.9 29.2 68328 5.98 3.40 0.07911 0.773 1.022 369 48.6 78.7 28.2 29.2 73174 5.75 3.29 0.08312 0.760 1.022 366 49.1 75.0 28.2 29.6 68180 5.66 3.20 0.08313 0.773 1.022 398 51.5 75.2 28.0 30.8 71894 5.89 3.32 0.08414 0.824 1.023 369 51.7 80.0 29.0 28.8 72490 6.05 3.46 0.07915 0.770 1.023 377 50.1 76.3 29.3 30.1 70595 5.82 3.25 0.08216 0.762 1.023 368 48.2 79.3 29.2 29.2 72185 5.97 3.35 0.08417 0.775 1.024 364 50.0 80.0 27.9 30.6 71114 5.59 3.46 0.08518 0.755 1.024 388 47.9 75.3 28.9 30.7 68065 6.13 3.49 0.07819 0.795 1.025 365 49.1 78.8 26.6 29.0 69736 5.81 3.50 0.08520 0.780 1.025 385 49.6 79.5 27.5 30.7 72428 5.90 3.24 0.07921 0.800 1.025 372 47.7 75.2 27.6 28.7 71694 5.90 3.29 0.08222 0.771 1.025 398 50.6 79.1 27.3 31.2 69369 5.87 3.51 0.08123 0.805 1.026 376 49.6 79.3 26.8 28.6 70611 6.01 3.28 0.08424 0.759 1.025 376 48.3 80.0 27.8 30.8 73371 6.09 3.42 0.08525 0.774 1.025 395 48.0 79.2 28.6 29.8 72277 5.86 3.31 0.08026 0.791 1.026 378 50.8 78.0 27.1 31.1 70251 6.02 3.42 0.08327 0.791 1.026 379 50.5 75.3 29.2 30.6 69763 5.92 3.45 0.08128 0.754 1.025 370 49.8 79.4 27.0 29.3 73230 5.61 3.25 0.08229 0.801 1.026 381 50.2 78.7 28.0 29.9 71487 5.87 3.36 0.08130 0.756 1.026 375 52.7 78.8 27.2 28.7 71045 6.05 3.31 0.08131 0.795 1.026 397 48.0 76.6 29.2 30.9 68766 6.14 3.27 0.08532 0.807 1.027 400 51.9 79.3 27.7 28.8 74095 5.98 3.40 0.08233 0.787 1.027 373 52.4 79.4 29.1 29.1 70561 5.76 3.41 0.07834 0.800 1.027 383 49.6 76.0 27.7 31.0 70467 5.99 3.32 0.08535 0.756 1.026 387 51.9 77.7 29.3 31.0 68402 5.77 3.30 0.08036 0.761 1.027 395 51.1 78.0 27.5 30.7 73661 5.99 3.25 0.08537 0.786 1.027 368 51.7 79.4 29.4 29.6 73884 5.85 3.40 0.08338 0.763 1.027 386 50.6 76.5 28.2 29.4 72401 6.07 3.42 0.08139 0.779 1.027 365 48.8 76.9 27.3 29.9 73962 5.84 3.35 0.08440 0.750 1.027 368 47.8 75.5 27.6 28.4 69499 5.99 3.49 0.08241 0.778 1.028 340 56.1 77.4 26.5 29.2 69320 7.28 3.49 0.10942 0.782 1.028 366 50.8 79.7 29.0 30.8 68753 5.96 3.42 0.08543 0.797 1.028 395 50.1 79.3 28.2 30.7 68397 5.86 3.42 0.08144 0.772 1.028 394 48.2 78.7 27.4 29.9 73106 5.71 3.20 0.08545 0.807 1.028 379 52.1 79.2 28.2 29.6 72839 5.68 3.52 0.07946 0.782 1.028 383 50.6 79.9 27.4 30.3 68523 6.15 3.47 0.08247 0.769 1.028 398 48.3 75.3 28.1 29.4 69946 6.15 3.38 0.07848 0.802 1.029 381 60.6 78.5 27.0 27.4 69998 6.48 3.56 0.34449 0.771 1.028 389 48.6 78.4 29.1 28.7 68568 5.94 3.51 0.08450 0.830 1.029 386 51.8 79.8 27.1 28.7 73572 5.78 3.33 0.084

Values are ranked by relative distance to the nominal fitness point (1,1). Relative distances are Euclidean distances after normalizing for differences in scalesbetween the axes. EMG, electromyography; SD, standard deviation. nMU, number of motor neurons; rIN, range in innervation numbers; Rr, recruitment range;PDR1, peak discharge rate 1st recruited neuron; PDRn, peak discharge rate last recruited neuron; nMF, number of muscle fibers; FL, fiber length; CV, meanconduction velocity; CVSD, conduction velocity spread.

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2005), and recruitment range and peak discharge rates (Fugl-evand et al. 1993; Milner-Brown and Stein 1975). By perform-ing an exhaustive multivariate analysis that compared modeledand experimental EMG/force and force/force-variability rela-tions, we obtain a true indication of the capabilities of thismodel structure as implemented in today’s literature. A list of50 valid parameter sets is provided in Table 2, rank ordered bytheir proximity to the nominal fitness point (1,1). Given thecurrent model structure, these values can be used as startingpoints to reproduce functionally realistic experimental data. Aswith many complex models, however, caution is warrantedbecause the fitness of the model lacks robustness to experi-mentally realistic parameter variability. Simulations in thevicinity of these parameter sets are not guaranteed to be asexperimentally valid (e.g., Fig. 5, inset).

This study suggests specific computational means to im-prove our understanding and representation of the mechanismsof motor-unit function. The results suggest reevaluating thegain functions describing the scaling of motor-unit dischargerate with excitation and the scaling of motor-unit force withdischarge rate because both gain functions are influenced bypeak discharge rates and recruitment ranges. The first gainfunction in the model assumes a linear scaling of motor-unitdischarge rate with increased excitation (Kernell 1965;Schwindt and Calvin 1972), although important nonlinearitiesare known to exist (Hultborn et al. 2003; Kernell 1965). Thesecond gain function assumes a sigmoid relation betweenmotor-unit force and discharge rate (Rack and Westbury 1969),normalized to the contractile properties of the motor unit(Kernell et al. 1983), though motor-unit force and contractiontime are not always correlated (Bigland-Ritchie et al. 1998).Moreover there is likely substantial context-dependent vari-ability in the scaling described by both gain functions acrossmotor units, which has not been implemented in the model.Also additional features not previously implemented in themodel could be included to test whether its experimentalvalidity improves. First, excitatory drive is not constant (ascommonly modeled) but may vary systematically with increas-ing contraction intensity, for example, due to signal-dependentnoise (Harris and Wolpert 1998). Thus an increase in variabil-ity of excitatory drive with increasing contraction intensitymay attenuate the unrealistic drop-off found in variability offorce at high force levels (Fig. 2A). Second, there may be apreferential distribution of low-threshold motor units deepin some muscles, inferred from the finding that slow, oxi-dative muscle fibers are more densely represented distantfrom the skin in some human muscles (Johnson et al. 1973;Lexell et al. 1983; cf. Edgerton et al. 1975). Thus at lowlevels of excitation, more active muscle fibers may besituated deeper in the muscle and thereby contribute less tothe surface-detected EMG signal, potentially attenuating theunrealistic increase in EMG at low-to-moderate levels offorce (Fig. 2B).

This work suggests iterations between modeling and exper-imental work will improve our understanding of the mecha-nisms of muscle function. For example, the feasibility ofisolating a small number of axons from the motor neuron poolto a muscle in situ has been demonstrated (Day and Hulliger2001; Perreault et al. 2003). This holds the promise of allowingthe delivery of known activation patterns (e.g., specific recruit-ment and rate coding schemes) simulating voluntary contrac-

tions to validate the modeled gain functions for the generationof EMG and muscle force. In addition, alternative “muscleoutputs” can be used as fitness criteria depending on thefeatures of muscle function being investigated, such as powerspectral densities. The present study is therefore an example ofhow computational modeling of neuromuscular systems coulddirect future experimental work to better understand the mech-anisms that give rise to muscle function and EMG.

A C K N O W L E D G M E N T S

We thank Drs. Madhusudhan Venkadesan and Veronica J. Santos for helpfulcomments on the manuscript.

The contents of this work are solely the responsibility of the authors and donot necessarily represent the official views of the National Institute of Arthritisand Musculoskeletal and Skin Diseases (NIAMS), the National Institute ofChildhood and Human Development (NICHD), the National Institutes ofHealth, or the NSF.

G R A N T S

This work was supported by National Science Foundation Grant 0237258and National Institutes of Health Grants R21-HD-048566, R01-AR-050520,and R01-AR-052345.

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